Question

How well can we evaluate a regression equation “fits” the data by examining the R Square statistic, and test for statistical significance of each independent variable in the regression equation by using the t-test?

Answer 1

A particular regression involves two kinds of value for each data points:

- Observed/Actual Values

- Fitted Values

Residuals =  Observed/Actual Values - Fitted Values

The main objective of every regression model is to minimize the difference between fitted and actual values. The regression line passing all the data points actually tells us the minimization of this distance as much as possible.

Now, R2 which is also known as the coefficient of determination and the Goodness-of-Fit tells us the percentage of variation in dependent variable that is explained by the linear model.

It is the measure that tells us how close the actual values to the fitted values. A 0 R2 indicates no variability in the dependent variable is explained by the model. In this case, the residuals = Observed values - Fitted Values, tend to be very high. On the other hand, 100% R2 tells us the gap between fitted and observed values is 0 and 100% variation in dependent variable is explained by the model.

T-tests are usually conducted to test if the model parameters are statistically significant and different from zero. It helps the research in differentiating between the significant and non-significant variables. The significance is inherently linked to model fitness. If the researcher includes only the insignificant variables, then the model fitness could be very poor. It is in this sense that the t-test is important in terms of examining how the regression equation fits the data.

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